• 투고 : 2016.03.25
  • 발행 : 2017.03.31


Let $F{\subseteq}{\mathbb{P}}^3$ be a smooth surface of degree $3{\leq}d{\leq}9$ whose equation can be expressed as either the determinant of a $d{\times}d$ matrix of linear forms, or the pfaffian of a $(2d){\times}(2d)$ matrix of linear forms. In this paper we show that F supports families of dimension p of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large p.


  1. M. F. Atiyah, Vector bundles over an elliptic curves, Proc. Lond. Math. Soc. 7 (1957), 414-452.
  2. J. Backelin, J. Herzog, and H. Sanders, Matrix factorizations of homogeneous polynomials, in Algebra-some current trends (Varna, 1986), 1-33, L.N.M. 1352 Springer, 1988.
  3. A. Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39-64.
  4. A. Beauville, Ulrich bundles on abelian surfaces, Proc. Amer. Math. Soc. 144 (2016), 4609-4611.
  5. A. Beauville, Ulrich bundles on surfaces with $p_g\;=\;q\;=\;0$, arXiv:1607.00895 [math.AG].
  6. J. Brennan, J. Herzog, and B. Ulrich, Maximally generated Cohen-Macaulay modules, Math. Scand. 61 (1987), no. 2, 181-203.
  7. M. Casanellas and R. Hartshorne, ACM bundles on cubic surfaces, J. Eur. Math. Soc. 13 (2011), no. 3, 709-731.
  8. M. Casanellas, R. Hartshorne, F. Geiss, and F. O. Schreyer, Stable Ulrich bundles, Internat. J. Math. 23 (2012), no. 8, 1250083, 50 pp.
  9. G. Casnati, Rank 2 stable Ulrich bundles on anticanonically embedded surfaces, to appear in Bull. Aust. Math. Soc..
  10. E. Coskun, R. S. Kulkarni, and Y. Mustopa, Pfaffian quartic surfaces and representations of Clifford algebras, Doc. Math. 17 (2012), 1003-1028.
  11. E. Coskun, R. S. Kulkarni, and Y. Mustopa, The geometry of Ulrich bundles on del Pezzo surfaces, J. Algebra 375 (2013), 280-301.
  12. L. Costa and R. M. Miro-Roig, GL(V )-invariant Ulrich bundles on Grassmannians, Math. Ann. 361 (2015), no. 1-2, 443-457.
  13. L. Costa, R. M. Miro-Roig, and J. Pons-Llopis, The representation type of Segre varieties, Adv. Math. 230 (2012), no. 4-6, 1995-2013.
  14. Y. Drozd and G. M. Greuel, Tame and wild projective curves and classification of vector bundles, J. Algebra 246 (2001), no. 1, 1-54.
  15. D. Eisenbud and J. Herzog, The classification of homogeneous Cohen-Macaulay rings of finite representation type, Math. Ann. 280 (1988), no. 2, 347-352.
  16. D. Eisenbud, F. O. Schreyer, and J. Weyman, Resultants and Chow forms via exterior syzigies, J. Amer. Math. Soc. 16 (2003), no. 3, 537-579.
  17. D. Faenzi and J. Pons-Llopis, The CM representation type of projective varieties, arXiv:1504.03819 [math.AG].
  18. H. Grassmann, Die stereometrischen Gleichungen dritten grades, und die dadurch erzeugten Oberflachen, J. Reine Angew. Math. 49 (1855), 47-65.
  19. Ph. Griffiths and J. Harris, On the Noether-Lefschetz Theorem and some remarks on codimension-two cycles, Math. Ann. 271 (1985), no. 1, 31-51.
  20. R. Hartshorne, Algebraic Geometry, G.T.M. 52, Springer, 1977.
  21. G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. 14 (1964), 689-713.
  22. D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, Second edition, Cambridge Mathematical Library, Cambridge U.P., 2010.
  23. J. O. Kleppe and R. M. Miro-Roig, The representation type of determinantal varieties, Preprint.
  24. R. M. Miro-Roig and J. Pons-Llopis, Representation Type of Rational ACM Surfaces X ${\subseteq}{\mathbb{P}}^4$, Algebr. Represent. Theory 16 (2013), no. 4, 1135-1157.
  25. R. M. Miro-Roig and J. Pons-Llopis, N-dimensional Fano varieties of wild representation type, J. Pure Appl. Algebra 218 (2014), no. 10, 1867-1884.
  26. C. Okonek, M. Schneider, and H. Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, 3. Birkhauser, Boston, Mass., 1980.
  27. J. Pons-Llopis and F. Tonini, ACM bundles on del Pezzo surfaces, Matematiche (Catania) 64 (2009), no. 2, 177-211.
  28. B. Ulrich, Gorenstein rings and modules with high number of generators. Math. Z. 188 (1984), no. 1, 23-32.

피인용 문헌

  1. Special Ulrich bundles on non-special surfaces with pg = q = 0 vol.28, pp.08, 2017,